Switching Quantum Reference Frames for Quantum Measurement

Physical observation is made relative to a reference frame. A reference frame is essentially a quantum system given the universal validity of quantum mechanics. Thus, a quantum system must be described relative to a quantum reference frame (QRF) using relational observables. To address this requirement, two approaches are proposed in the literature. The first one is an operational approach (F. Giacomini, et al, Nat. Comm. 10:494, 2019) which focuses on the quantization of transformation between QRFs. The second approach attempts to derive the quantum transformation between QRFs from first principles (A. Vanrietvelde, et al, arXiv:1809.00556). Such first principle approach describes physical systems as symmetry induced constrained Hamiltonian systems. The Dirac quantization of such systems before removing redundancy is interpreted as perspective-neutral description. Then, a systematic redundancy reduction procedure is introduced to derive description from perspective of a QRF. The first principle approach recovers some of the results from the operational approach, but not yet include an important part of a quantum theory - the measurement theory. This paper is intended to bridge the gap. We show that the von Neumann quantum measurement theory can be embedded into the perspective-neutral framework. This allows us to successfully recover the results found in the operational approach, with the advantage that the transformation operator can be derived from the first principle. In addition, the formulation presented here reveals several interesting conceptual insights. For instance, the projection operation in measurement needs to be performed after redundancy reduction, and the projection operator must be transformed accordingly when switching QRFs. These results extend our understanding on how quantum systems should be described when the reference frame is also a quantum system.


I. INTRODUCTION
The idea that a physical system or a physical phenomenon must be described relative to a reference frame is a well-accepted principle in the relativity theory. Abandoning the concept of absolute spacetime is a foundation of the general relativity where the laws of physics is invariant when changing reference systems. A reference frame essentially is also a quantum system, if we agree that quantum mechanics is universally valid. Thus, a physical system or a physical phenomenon must be described relative to another quantum system. This statement is not only applied to describe a relativity event, but also applicable to descriptions of all quantum phenomena. The implication here is that a more fundamental theory should describe a physical system relative to a quantum reference frame (QRF), and address how such descriptions are transformed from one to another when switching the QRFs.
There are extensive research literature related to QRFs [1][2][3][4][5][6][7][8][9][10][11][12][13]. Here we are only interested to those theories that satisfy two criteria: 1.) Completely abandoning the concept of classical system. All reference systems are quantum systems instead of some kinds of abstract entities. Treating a reference frame as a classical system, such as how the relativity theory does, should be considered as an approximation of a more fundamental * jianhao.yang@alumni.utoronto.ca (preferred) theory that is based on QRF. 2.) Completely abandoning any external reference system and the concept of absolute state. Physical description is constructed using relational variables from the very beginning. These criteria are at the heart of the relational quantum mechanics [14][15][16][17][18][19] 1 .
Given these criteria, two approaches stand out as of importance. The first one is an operational approach [20] that focuses on how the dynamical physical laws are transformed when switching QRFs. By operational approach, we mean it assumes that every QRF is equipped with hypothetical measuring apparatus that can perform measurement to justify a state assignment. The advantage of such approach is that the formulation can be potentially tested in a laboratory. The theory has produced several interesting results. For instance, it allows us to derive the operational meaning of spin of a relativistic particle; It shows that entanglement among quantum systems depends on the choice of QRF; It also shows how the measurement outcomes from perspectives of two different QRFs can be related. The limitation of the operational approach is that it assumes the transformation between two QRFs is known beforehand. The operator associated with the transformation is derived primarily from intuition.
The second approach [21], which we call it a first principle approach, attempts to address the limitation in the operational approach. The starting point in this approach is the same, that is, quantum systems should be described with relational variables only. What is novel in this approach is that it implements such idea using the tools and concepts of constraint Hamilton systems [22,23] 2 . The constraint conditions encode the corresponding gauge symmetries. For instance, the translational symmetry results in a constraint that total momentum to be zero. This implies the system should be described in a constraint surface in the phase space. There are two commonly used methods to canonically quantize the constrained systems, namely, the reduced quantization, and the Dirac quantization. In the reduced quantization method, one solves the constraints first in the context of classical mechanics, then quantizes the reduced theory. Solving the constraints is interpreted as taking a perspective of a particular quantum reference frame. Thus this method derives the quantum theory directly for a particular quantum reference frame. On the other hand, in the Dirac quantization method, one quantizes the system first without considering the constraints, then solves the constraints in the quantum theory. The quantized theory before removing the symmetry related redundancy is interpreted as a perspective-neutral theory and it does not admit an operational interpretation. When the redundancy is removed, the theory is reduced to the perspective from a particular QRF. The reduced theory then admits operation interpretation. Removing the symmetry is equivalent to fixing the gauge in the case of classical mechanics. But in the case of quantum mechanics, there is subtlety that there is no globally valid gauge fixing condition in some generic systems such as the N-body three dimensional systems [26]. Regardless of the interpretation, the essence of the procedure is to remove the symmetry related redundancy after the Dirac quantization. The procedure to ensures that the Dirac quantization theory is consistently mapped to the reduced quantization theory. The procedure can be applied to different QRFs, resulting reduced quantum theory for different QRFs, this allows us to derive the transformation formulation between QRFs.
The first principle approach has successfully recovered some of the results in the operational approach, with the advantage that the transformation operators are naturally derived. These include the quantum state transformation when switching QRFs, the Schrodinger equation, and the conclusion that entanglement among quantum systems depends on the chosen QRF. However, an important part of a quantum theory, the measurement theory, is not yet studied.
The goal of this paper is to investigate how the measurement theory can be embedded into the first-principle approach and how to recover the results related to measurements in the operational approach. We focus on the von Neumann quantum measurement theory [27], where the measurement stages are separated into two sub processes, namely, Process 1 and Process 2. In Process 2, the measured system and the measuring apparatus interact, but the combined systems can be described as unitary process and determined by the Schrodinger Equation. At the end of the interaction, the combined systems are entangled, and multiple outcomes are possible. Each of the outcome is associated with a specific value of a pointer variable of the apparatus and a probability. Then, in Process 1, which is described by a projection operation, a specific outcome is singled out. We will show that Process 2 can be described in the perspective-neutral framework, while Process 1 in general needs to be described with respect to a particular QRF. When the pointer variable is invariant in the symmetry reduction procedure, it is mathematically equivalent to describe Process 1 in the perspective-neutral framework as well.
In Section III, we show how the von Neumann measurement theory can be successfully embedded into the perspective-neutral framework. We are able to derive the measurement formulation relative different QRFs that are consistent with the results in the operational approach [20]. Furthermore, our results reveal additional insights on the measurement theory when switching QRFs. For instance, 1.) It is crucial to consider the order of operations when formulating the measurement theory: should we perform the quantum symmetry reduction procedure first, or projection first? 2.) When switching QRFs, the projection operator needs to be transformed properly in order to preserve the measurement probability. This second point fine-tunes the synchronization principle proposed in Ref. [19,34] to resolve the extended Wigner's friend paradox. The implications of these conceptual results are further discussed in Section IV. In conclusion, together with Ref. [20,21], we extend the understanding of how quantum mechanics is formulated when the reference frame itself is a quantum system and when the switching QRFs.

II. SWITCHING QRFS VIA A PERSPECTIVE-NEUTRAL FRAMEWORKS
This section briefly reviews the first principle approach developed in Ref. [21] for switching QRFs via a perspective-neutral framework. For convenience, we will adopt the same notations used in Ref. [21]. At the core of the theory is the notion that physics is purely relational. Physically meaningful variables are relational observable, they are invariant under certain gauge transformation. To achieve such description, the systems under studied are described as constrained Hamiltonian systems [22,23].

A. A Toy Model
The theory is illustrated through a simple one dimensional toy model. We start to describe the toy model in the context of classical mechanics. The model consists N particles with unit mass and canonical pairs {p i , q i }. These canonical pairs define a 2N phase space. To ensure only relational observables are used to describe the systems, the Lagrangian of such systems, in the context of classical mechanics, must be invariant under global translation. This requirement leads to the following constraint i.e., the momenta of the particles are not all independent. The constraint reduces the phase space dimension to be 2N − 1. The above equation thus defines a constraint surface in the original phase space such that it only holds in that constraint surface (the symbol ≈ is used to reflect this weak equality). A Dirac observable O is defined as a variable that is Poisson-commuting with P in the constraint surface, i.e., {O, P } ≈ 0. Obviously all momenta p i and the relative distance q i − q j are all Dirac observables. The total Hamiltonian is read as [22,23] H tot = 1 2 where λ is the Lagrangian multiplier. It is determined when choosing a reference frame thus fixing the gauge. For instance, if one chooses particle A as reference frame, the gauge is fixed, and it is defined as choosing q A = 0 in the constraint surface. The momentum of A is solved from (1): From the equations of motion and the gauge fixing condition q A = 0, one can derive that λ = −p A . Then, the corresponding reduced Hamilton from A perspective can be computed as Now we quantize this toy model. For simplicity, only three particles A, B, C, are considered, i.e., N = 3. First we consider the reduced quantization method. Supposed A is chosen as QRF, we need to quantize the theory in the reduced phase space from A's perspective. The standard procedure is to promote {p i , q i , i = A} to operators that satisfy the following commutation relations: This defines a 2(N − 1) dimensional Hilbert space. In the N = 3 case, the reduced Hamiltonian (4) is quantized aŝ An arbitrary state vector for particle B and C, from A's perspective, can be written as

B. Dirac Quantization
In the Dirac Quantization method, one first quantizes the system without considering the constraints. This is achieved by promoting all {p i , q i } to operators and with appropriate commutation relations. This defines a 2N dimensional kinematical Hilbert space, H kin . Next, the momentum constraint (1) is quantized in this Hilbert space. In the Dirac quantization method, this is amounted to require that the physical states of the systems are annihilated by the momentum constraint operator, i.e.,P |ψ phys = (p A +p B +p C )|ψ phys = 0.
To ensure proper inner product is well-defined for the physical state, a physical Hilbert space H phys is constructed from H kin by an improper projector δ(P ) : H kin → H phys . This is defined by the following map [25], With this definition, from an arbitrary state in H kin , one can obtain the solution for |ψ phys , if solving the constraint for particle A, as It can be verified that (8) is satisfied. The sought-after proper inner product for |ψ phys is (ψ phys , φ phys ) = ψ|δ(P )|φ kin .
So far the quantization procedure follows the standard Dirac quantization for constraint system. The novelty in Ref [21] is to interpret the results of the Dirac quantization as a perspective-neutral framework for the systems. It takes all perspectives of reference systems at once, be it from particle A, or B, or C. To recover the relative states from Dirac quantization to a reduced quantum theory for a specific reference frame, Ref [21] proposes the following procedure: Step 1, pick a reference system, say, particle A. The physical state is then given in (11).
Step 2, apply a unitary transformation defined in H kin This effectively defines a new representation of the physical Hilbert space. The physical state (11) becomes It is important to note that such transformation is a global operation acting on all three particle at once instead of just a local operation acting on only one particle. Thus, it can change the entanglement properties among the particles.
Step 3, apply a projection to remove the redundancy of the degree of freedom from the reference system. This also eliminates the self-reference problem. The reduced state from reference system A, is This recovers the same reduced state as in (7). The resulting Hilbert space is denoted as H BC|A .

C. Switching QRFs
If we repeat the three steps in the previous subsection but picking particle C as the reference system, we obtain the reduced state , and the resulting Hilbert space is denoted as H AB|C . Switching QRF from particle A to C is described by the following map:Ŝ Such map can be derived by inverting the transformation from A-perspective back to the neutral-perspective framework and then applying the transformation to Cperspective. It connects the two reduced states asŜ A→C |ψ BC|A = |ψ AB|C . Ref [21] shows that this map is equivalent toŜ whereP CA is the parity-swap operator 3 , which when acting on momentum eigenstate of C yieldŝ Thus, the theory developed in Ref [21] is equivalent to that in Ref [20]. This concludes the overview of the first principle approach to switch QRFs via the perspective-neutral framework. The goal of this paper is to extend this theory to the quantum measurement formulation. We wish to develop the theory for quantum measurement starting from Dirac quantization, and follow the same symmetry reduction procedure to derive the reduced theory for a specific reference frame. It is also expected that by swapping between two QRFs, the measurement theory consistently recovers the results in the operational approach.

III. QUANTUM MEASUREMENT
To investigate how the quantum measurement process be embedded in a perspective-neutral framework, we employ the von Neumann quantum measurement theory, where the measurement stages are separated into two sub processes, Process 1 and Process 2 as mentioned in the introduction section. It is pointed out [19] that Process 1 must be described explicitly as observer-dependent 4 . Different observers may have different descriptions of the same measurement event, which is vividly manifested by the Wigner's Friend thought experiment [28,29]. An observer is associated with a specific QRF, this means Process 1 must be described specifically relative to a QRF. This is consistent with the fact that Process 1 is operational. An operational process should not be described in the perspective-neutral framework as the perspectiveneutral framework does not admit operational interpretation [21]. Therefore, Process 1 needs to be described in the reduced Hilbert space, e.g., H AB|C if particle C is the chosen QRF. On the other hand, Process 2 can be described in the perspective-neutral framework, i.e., in the Hilbert space H phys .
With these guidelines in mind, our strategy is to formulate Process 2 in H phys , perform the redundancy reduction (or, symmetry reduction) procedure, and finally apply the projection operator representing Process 1 in the reduced Hilbert space.

A. Measurement Theory for the Toy Model
The same toy model is used to develop the measurement theory. Besides particle A, B and C, another particle E is added as a measuring (or, auxiliary) particle. In this setup, we assume there is a pointer variable for particle E that is used to measure particles A and B. For simplicity, the momentum of particle E is chosen as pointer variable. We will derive the measurement formulation by initially choosing particle C as the QRF, then examine the theory by switching QRF from particle C to particle A.
The momentum constraint (8) is extended to include particle E, so that The measurement formulation typically starts with the assumption that the initial state is a product state between the measurement apparatus system and the rest of system. In H kin , the initial state is written as where we assume E is in an eigenstate with momentum p 0 . After applied the constraint map δ(P ), the physical state for the systems can be written The Process 2, where particle E interacts with the rest of system and becomes entangled with the rest of system, is described as unitary process in the von Neumann measurement theory. Denote the unitary operator in H phys asÛ = e −iĤt , whereĤ is the total Hamiltonian in H phys that includes an interaction term between E and the measured particles. With these notations, Process 2 is formulated asÛ whereΛ pE = E p E |Û |p 0 E is an operator defined in H phys and only acts on particle A, B, and C. Note that there are infinite number ofΛ pE and they are labeled with index p E . Now we start the symmetry reduction procedure by following the three steps described in previous section. First, particle C is picked as the reference frame. Next, a transformation operator, defined below 5 , 5 As pointed out in Ref [21], the transformation operatorT C is not unique. A constant p 0 can be added inside the parentheses and will not impact the relevant information for particles A, B, and E that are involved in the measurement process.
is applied to (23), Insert a unitary operatorT 0 = e iqC (pA+pB +p0) and denotê T ′ = e iqC (pA+pB +pE ) , we rewrite (25) aŝ where where operatorΓ pe only acts on particle A and B and is defined in (A5). Substitute this into (26), we find TÛ |ψ phys = |p = 0 C ⊗ dp EΓpE |φ C,AB |p E E . (29) As the final step of the reduction procedure, we discard the redundant state for particle C.
At this point, the pointer variable p E is well-defined in the reduce Hilber space H ABE|C . The Process 1 in von Neumann measurement theory is described by applying the projectionP to (30), resulting Dropping the state for particle E, and definingM m = Γ pm / √ N , where N is a normalization factor, we further rewrite the state vector for particle A and B after measurement as the well-known form, The probability of the measurement outcome for m is The full expression of operatorM m , as derived fromÛ , is expressed aŝ Appendix C verifies that the set of operators {M m } satisfy the completeness condition, i.e., whereÎ AB|C is a unit operator in Hilbert space H AB|C .

B. Switching QRFs
Next we switch the QRF to be particle A and examine how the measurement formulation changes. Following the same derivation procedure, we find the final state for particle B and C after measurement is where From (33) and (37), we can derive how the two state vector |φ m BC|A and |φ m AB|C are connected. Define a transformation operator whereP AC is the parity swap operator defined in (19).
In Appendix B, we prove that whereŜ m =P AC e iqA(pB +pm) = E p m |Ŝ|p m E . As also shown in Appendix B, the probability of measurement outcome m from the perspective of QRF A is conserved, i.e., λ m|A = λ m|C . The conservation of the probabilities of the same measurement outcome from perspectives of both QRFs is a natural consequence of the formulation, owning to the unitary property of operator S m . This result is consistent with that in Ref [20].
It it important to note that the derivation of (37) depends on two assumptions. First, the observer associated with QRF A knows the same measurement outcome m as the observer associated with QRF C. This should not be taken for granted if A and C are separated remotely. The synchronization of the information regarding the measurement outcome is necessary [19]. Second, the pointer variable projectionP m , defined in (31), is invariant under the transformation of QRF. One can easily verify that To demonstrate the effect ofŜ m , we consider a concrete example. Supposed the measurement operatorM m projects the state for particle A and B into a product state from C perspective, i.e., The final state after measurement, according to (33), is Thus, from C perspective, A, B, and E are not entangled with each other after measurement. Now if we switch the QRF from C to A, from A perspective, the observed probability and the value of pointer variable corresponding to m are the same, but the quantum state for B and C after measurement is It shows that B and C are entangled. This result is similar to that in Ref [20], except that the auxiliary system E is not necessarily entangled with B or C.

C. Measurement Apparatus as QRF
So far the formulation assumes the measuring system (or, auxiliary system) and the QRF are two different systems. However, there is an important situation where we need to treat the measuring system itself as the QRF. For instance, supposed system A is the laboratory, systems B and C are moving with a same speed relative to A, and C is measuring B. It is more difficult to describe the measurement process from A perspective, particularly when the pointer variable depends on the speed. It is much easier to description the measurement process first at the rest reference frame and then transform to describe the process from A perspective. In this case, it means we take the measuring system C also as the QRF, derive the measurement formulation from C perspective, then transform back to the perspective of A as QRF 6 .
In Ref. [20], such scenario is considered where system C has both external degree of freedom (i.e., the momentum degree of freedom) that is discarded when A is taken as a QRF, and the internal degree of freedom that is acting as a pointer variable to measure system B.
In the toy model considered in this paper, particles A, B, and C are interacting and only constrained with a global translational invariance. Relative speed is not considered in this simple model. We assume particle C has both external degree of freedom, and internal degree of freedom that acts as a pointer variable. Thus, a general state for particle C, before considering the constraint, is An eigen state associated with a particular value of pointer variable σ m = σ n is The state vector satisfies the following orthogonal identities: The pointer variable (i.e., the internal degree of freedom), which can be the energy level, or the spin, in theory can depend on the momentum variable. A good example is the spin of a relativistic particle can depend on the velocity of the particle. For simplicity we assume here that the pointer variable is independent of the momentum variable. Similar to (21), we assume the initial state of A, B, and C is a product state in H kin , written as The Process 2, where particle C interacts with B, is described as a unitary process. The unitary operatorÛ kin is associated with the HamiltonianĤ kin defined in H kin .
whereΓ m = C φ m |Û |φ 0 C is an operator defined in H kin and only acts on particle A and B. Next we apply the constraint map δ(P ) to (49). On the left hand side, the unitary operatorÛ kin changes to a unitary opera-torÛ phys where the correspondent Hamiltonian isĤ phys . Similarly, |ψ phys = δ(P )|ψ kin . We hereafter omit the superscript phys of the operators for simple notation. Thus, in Hilbert space H phys , (49) is changed tô To proceed with the redundancy reduction procedure, particle C is picked as the reference frame. The transformation operatorT C is simply defined below 7 In Appendix D, we show that after applyingT C to (50), we obtain where operatorM m is defined in (D5). The last step of removing the redundancy of external degree of freedom of QRF C can be combined with the projection of the Process 1 of the measurement. Namely, for a measurement outcome corresponding to pointer variable of σ n , we can apply C p = 0, σ n |⊗ to (52), and obtain the final state after the measurement The more interesting question is how the same measurement process is perceived if we choose particle A as QRF. In Appendix E, we show that starting from the same decomposition (49), after carrying over the redundancy reduction procedure with particle A as the QRF, and with transformation operator defined asT A = e iqA(pB +pC ) , we get where χ m (p B , p C ) is defined in (E3). The question here is that what the projection operator for Process 1 we should apply at this point. Since we assume the pointer variable does not transform under translation, from A perspective, the pointer variable is the same as from C perspective. Thus, we define the projection operator from C perspective asP n = |σ n C σ n |, assuming that the pointer variable σ m is separable from the momentum degree of freedom p C , i.e., |p C , σ m = |p C |σ m . ApplyingP n to (54), we obtain the final state after measurement, (55) shows that in the final state from A perspective, the degree of freedom of particle B is entangled with external degree of freedom of C, but disentangled with the internal degree of freedom of C, i.e., the pointer variable of C. This reproduces the same conclusion in Ref [20]. Defining transformationŜ n =P n AC e iqApB wherê P n AC |p A = | − p, σ n C . (56) We show in Appendix F that (55) and (53) can be transformed similar to (40) It follows immediately that the probability of measurement outcome correspondent to pointer variable value σ n is the preserved from the perspective of either QRF.

A. The Order of Reduction and Projection
The projection in the Process 1 of the von Neumann quantum measurement theory is an operational process, i.e., a process that can be confirmed or observed through the pointer variable of the measurement apparatus. The value of the pointer variable is read and recorded with respect to a specific QRF. Thus, it should be described in the reduced Hilbert space with respect to that particular QRF. This means we perform the symmetry reduction of the Dirac quantization first, then the projection operation. As a consequence, the operational meaning of the pointer variable is well-defined in the reduced Hilbert space. Such a procedure is consistent with the expectation that a measurement event should be operationally well-defined. An opposite procedure is to perform the projection first in the perspective-neutral Hilbert space, then the symmetry reduction procedure. However, the projection operator can be transformed during the reduction procedure, and the operational meaning of that pointer variable becomes difficult to define. For instance, let's consider particle A, B, C are relativistic particles with spins and moving with different speeds. Supposed the spin of particle C is the pointer variable for a measurement on particle B and C itself is also the QRF. The operation meaning of spin is well defined in a rest QRF with particle C. On the other hand, it is a challenge to perform the projection operation in the perspectiveneutral framework. This is because the spin of particle C depends on its momentum and particle C can be in a superposition state of momentum. The symmetry reduction procedure involves transformation of the momentum operator which in turn transforms the spin. Thus, it is not clear how to define the projection operator in the perspective-neutral framework 8 .
There is, however, a special situation when the pointer variable is invariant in the reduction procedure. In this case, the order of reduction and projection does not impact the formulation of measurement. Considered the example in Section III C, where we perform reduction first, and projection later to derive the measurement formulation. But in fact the order can be reserved and yield the same results. Supposed we first apply the projection operatorP ′ n = |φ n C φ n | to (49), and obtain P ′ nÛ kin |ψ kin =Γ kin n |ϕ AB |φ n C .
Then we proceed the reduction procedure by choosing C as QRF, and obtain (53). If choosing A as QRF, the reduction process resulting with |ϕ n BC|A = dp B dp C φ n (p C )χ n (p B , p C )|p B |p C , σ n , (59) which is essentially the same as (55). The reason for this is that the pointer variable, σ m , is invariant in the symmetry reduction procedure, and consequently also invariant when switching QRFs. However, this is a special case and should not be considered true in general.

B. Synchronization Among Different QRFs
When deriving (37) or (55), we assume that the observer associated with QRF A knows the measurement outcome that is inferred from the pointer variable σ n associated with QRF C. In other words, we assume that the measurement projection operatorP n is known to both observers. This assumption should not be taken for granted. Quantum measurement must be described relative to the local observer. An observer who does not access to the measurement results will not have the complete information and can only describe the system up to the level of previous knowledge that the observer has. To ensure the descriptions of different observers are consistent, different observers should synchronize information regarding the measurement results in order to have consistent descriptions of a quantum system [19,34]. The results in this paper shows that a further refinement on the information synchronization is needed. Specifically, the pointer variable needs to be transformed properly when switching QRFs. In the special case that the pointer variable, consequently the projection operatorP n , are invariant from either A perspective or C perspective, we simply use the same operatorP n for the projecting operation. This is what we have done when deriving (37) or (57).
are moving with a speed relative to each other. However, it is not clear how the transformation operator from the perspectiveneutral Hilbert space to the reduced Hilbert space is defined.
To illustrate this point more concretely, we go back to the example in Section III C where particle C is the measurement apparatus and we choose particle A as QRF. After the symmetry reduction procedure we arrive at (54) and need to apply a projection operator for particle C from A perspective. Now, instead of choosinĝ P n|A = |σ n C σ c |, one may attempt to assign the projection operatorP ′ n|A = |φ n C φ n |. One reason this choice seems make sense is that an arbitrary wave function for particle C that is associated with pointer variable σ n is given by (46). Applying this projection operator to (54), we obtain a final state after measurement, Obviously, |ϕ ′ n BC|A is a product state. Particle B is not entangled with either the external degree of freedom or the internal pointer variable of particle C. This result is different from (55) where particle B is entangled with the external degree of freedom of C. Further calculation shows that the probability of the measurement resulting fromP ′ n|A is not the same as the probability from C perspective. The preservation of the measurement probability from different QRFs appears to be a natural criteria that should be satisfied. Thus, the choice of projection operatorP ′ n|A is incorrect. This observation can be further explained as following. If from C perspective the projection operator iŝ P n|C = |σ n C σ c |, then from A perspective, the corresponding projection operator must be derived via proper transformation,P n|A =Ŝ n (P n|C )Ŝ † n whereŜ n is defined in (56). SinceŜ n has no effect on |σ n C σ c |, we obtain P n|A =P n|C . This rules outP ′ n|A as the the correct projection operator.
We can summarize the implications of the relational formulation of quantum measurement based on Refs [19,34] and this paper as following.
1. Measured reality is relative. Information obtained through quantum measurement is local. Measurement must be described explicitly relative to the local observer.
2. Synchronization of local reality. Different observers should synchronize information regarding the measurement results in order to have consistent descriptions of a quantum system.
3. When the measurement information being synchronized among different observers, proper transformation on the projection operator must be performed to ensure consistency.
Applying the first implication, we are able to resolve the EPR paradox [19,32]. In that resolution, a quantum measurement should be explicitly described as observer dependent. The idea of observer-independent quantum state is abandoned since it depends on the assumption of Super Observer. By recognizing that the element of physical reality obtained from local measurement is only valid relatively to the local observer, the completeness of quantum mechanics and locality can coexist [19]. Applying the second implication, we are able to resolve the Wigner's friend paradox and the extended version [33,34]. These thought experiments provide clear example for the need of information synchronization in order to achieve a consistent description of a quantum system by different observers. Ref [35] shows similar idea that the assumption of observer independent fact cannot resolve the Wigner's friend type of paradox. Latest experiment appears to confirm that observer-independent description of a quantum system must be rejected [31]. Lastly, the third implication is a refinement of the second implication, and is just discussed in this section.

C. Limitation
The toy model used in this paper is relatively simply. The systems in this model are one dimensional and only have constraint due to the translational symmetry. The simple model allows one to construct a globally valid gauge fixing condition. Ref. [26] extends the method to three dimensional N-body systems that have both translational and rotational symmetries. It will be interesting to confirm that the measurement theory developed in this paper will be applicable to the three dimensional N-body systems as well. However, as Ref. [26] points out, for three dimensional N-body systems that have both translational and rotational symmetries, one cannot find globally valid gauge fixing conditions. As a consequence, there is no globally valid internal perspective from a particular system. This imposes a difficulty to interpret the symmetry reduction procedure from perspective neutral framework to QRF specific framework as a gauge fixing procedure. It seems it is more prudent to interpret the three-steps mathematical procedure described just simply as a redundant reduction procedure and not necessarily tied with gauge fixing. Regardless the interpretation, we expect that the measurement formulation presented in this paper is applicable to three dimensional N-body systems based on the formulation in Ref. [26]. It is desirable to extend the theory to even more complicated model that includes other degree of freedom such as spin.
The assumption that the pointer variable is invariant during the reduction process is another major limitation. If the pointer variable changes when switching QRFs or during the reduction process, the transformation operator such asT C needs to include the pointer variable, and the formulation will be more complicated. We would also expect that such condition should manifest the importance of the order of reduction versus projection, as discussed in Section IV A.
The projection process in the von Neumann quantum measurement theory is a simple mathematical modeling of the actual measuring process. It cannot explain the mechanism of "wave function collapse". Ref. [21] speculates that by including the measurement interaction into the perspective-neutral structure, it may possibly lead to the "collapse" in the respective internal perspective. Obviously our formulation here does not achieve such goal. However, we do bring in new conceptual implications discussed earlier.

D. Conclusions
Inspired by the novel approach of switching QRFs via a perspective-neutral framework [21], this paper extends the approach to the quantum measurement process. Specifically, we show the von Neumann quantum measurement theory can be embedded in the perspectiveneutral framework. Based on the same simple toy model as in Ref. [21], we show that Process 2 in the von Neumann measurement theory, which is a unitary process, can be formulated in the perspective-neutral framework, while Process 1, which is a projection, should be described after the perspective-neutral structure is reduced to be specifically relative to a QRF. In the special case when the pointer variable is invariant to the redundancy reduction procedure (hence invariant when switch-ing QRFs), the order of reduction versus projection has no impact on the results. Our results successfully reconstruct the measurement formulation, as shown in Section III A and III B, from perspectives of different QRFs. This allows us to further derive the transformation operator for the measurement outcome when switching QRFs, given in (39). These results are consistent with that in Ref. [20], with the advantage of being derived from the first principles proposed in Ref. [21].
Furthermore, when the measurement apparatus itself is considered as a QRF, our measurement formulation provides additional conceptual implications. In particularly, when switching QRFs, the projection operator in the must be transformed properly, otherwise it causes inconsistency. For instance, the probabilities from different observers for the same measurement outcome can be different. Conceptually, this further refines the synchronization principle [19,34] that is used to resolve paradox for the extended Wigner's friend thought experiment.
In conclusion, the results presented in this paper further confirm the validity of the first principle approach of switching QRFs via a perspective-neutral framework [21]. Our formulations on the measurement theory fills in the gap to recover the transformation mechanism when switching QRFs using the operational approach [20]. All these research results together extend the understanding on how quantum systems should be described when the reference frame itself is a quantum system and when the switching QRFs.
In general, the unitary operatorÛ can be decomposed asÛ = dp A dp B dp C dp E dp ′ A dp ′ B dp ′ C dp ′ Given the momentum constraint in Hilbert space H phys , the decomposition should be updated tô where u is a shorten notation for function . Thus, the decomposition of operatorΛ pE readŝ Now applying operatorT ′ andT 0 , we obtain where operatorΓ pE only acts on particles A and B, defined aŝ Γ pE = dp A dp B dp ′ A dp ′ Appendix B: Proof of Eq.(40) First we insert (39) into the following expression : whereŜ 0 =P AC e iqA(pB +p0) . From (27) and Appendix E of Ref. [21], we identifŷ S 0 |φ AB|C = |φ BC|A . (B2) Next step is to evaluateŜ mMmŜ † 0 . Note thatM m = Γ m / √ N . Substituting (A5) forΓ m , we havê S mMmŜ † 0 = 1 √ NP AC e iqA(pB +pm) × dp A dp B dp ′ A dp ′ B × u|p A |p B p ′ A | p ′ B | × e −iqA(pB +p0)P AC = 1 √ NP AC dp A dp B dp ′ A dp ′ B × u m

× |p
where u m is a shorten notation for function . Applied the change of variable −p C = p A + p B + p m and −p ′ C = p A + p B + p 0 , function u m becomes u(−p B − p C − p m , p B , p C , p m ; −p ′ B − p ′ C − p 0 , p ′ B , p ′ C , p 0 ), and (B3) can be rewritten aŝ S mMmŜ † 0 = 1 √ NP AC dp C dp B dp ′ C dp ′ B × u m × | − p C A |p B B A −p ′ C | B p ′ B |P AC = 1 √ N dp C dp B dp ′ C dp ′ B × u m |p C C |p B B C p ′ C | B p ′ B |.
One immediately recognizes the right hand side of (B4) is nothing butK m . Thus, Inserting (B2) and (B5) back into (B1), we obtain Applying E p m |⊗ on both sides of (B6), we get This implies that Given (B7) and (B8), and the definitions in (33) and (37), we obtain the desired identity (40).

Appendix C: Completeness of {Mm}
Note that index m refers to p m which is our toy model is actually a continuous real number, p m ∈ R. The completeness should read dp EM † pEMpE =Î AB|C . (C1) To verify that, we take the complex conjugate of (29), multiply to itself, and evaluation both sides: L.H.S. = phys ψ|Û †T †TÛ |ψ phys = phys ψ|ψ phys = kin ψ|δ(P )|ψ kin = N L .